We report on full-polarization radio observations of the Perseus cluster (Abell 426) using the Westerbork Synthesis Radio Telescope (WSRT) at wavelengths from 81-95 cm. We detect faint, very extended polarized emission throughout the cluster region. We have employed a novel technique, Rotation Measure synthesis (Brentjens & de Bruyn, 2005, A& A, 441, 1217) to unravel the polarization properties of the emission across the full field of view. We detect polarized emission over a wide range of RM from about 0 to 90 radm(-2). Low RM emission (RM 30 rad m(-2) on the other hand, shows organized structures on scales up to a degree and displays rapidly fluctuating polarization angles on scales of the synthesized beam. A Galactic foreground interpretation for the high RM emission can not be ruled out, but appears extremely implausible. WSRT observations at 21 cm of the RM of a dozen discrete sources surrounding the Perseus cluster indicate a smooth large-scale gradient in the Galactic foreground RM. The diffuse structures have a clear excess RM of about 40 rad m(-2) relative to these distant radio galaxies. This excess Faraday depth, the generally good spatial association with the cluster and the different morphology of the high RM emission, compared to the genuine Galactic foreground emission, all point to an association of the high RM emission with the Perseus cluster. The polarized emission reaches typical surface brightness levels of 0.5-1 mJy per 2' x 3' beam and must be rather highly polarized (greater than or similar to 20%). Due to dynamic range limitations and lack of sensitivity to large-scale structure we have not yet detected the corresponding total intensity. Most of the polarized emission, located at distances of about 1 degrees from the cluster centre, appears too bright, by about 1-2 orders of magnitude, to be explainable as Thomson scattered emission of the central radio source off the thermal electrons in the cluster. However, this remains a viable explanation for the highly polarized 21 cm emission from the inner 10'-20' and part of the 81-95 cm emission. The bulk of the emission associated with the Perseus cluster may instead be related to buoyant bubbles of relativistic plasma, probably relics from still active or now dormant AGN within the cluster. A lenticular shaped structure, referred to as the lens, and measuring 0.5-1 Mpc is strikingly similar to the structures predicted b En beta lin et al. ( 1998, A& A, 332, 395). At the western edge of the cluster, we detect very long, linear structures that may be related to shocks caused by infall of gas into the Perseus cluster along the Perseus-Pisces filamentary structure of the cosmic web.